July 9, 2020
Multi-dimensional optimization of a loudspeaker
Industry: Electronics | Software: pSeven | Company: Mvoid
Objective
An ideal loudspeaker translates a voltage signal via the excursion of a membrane into sound pressure by a linear dependency. The membrane is driven by a voice coil moving in the air gap of a magnet system. In a real system, the electrical properties of the voice coil depend on its position relative to the magnet system, and the magnetic flux density of the magnetic circuit is not constant over the area the voice coil moves. In the equivalent circuit, representing the behaviour of the transducer, the electrical components are not linear.
Transducer developers have many knobs to turn for getting closer to a linear behaviour over a wider excursion. Some knobs, however, have a negative impact on other parameters, e.g. a longer voice coil decreases the sensitivity. The goal is to find compromise between many objective functions by varying mechanical and electromagnetic characteristics of acoustic system, like magnet dimensions and tolerances, cone & surround mechanical properties and so on.
Figure 1. Principle schema of transducer & acoustic system.
Challenges
- Significant evaluation time of a single configuration characteristics comparing to total development lead time.
- High number of variables in consideration with different level of impact on the performance.
- Multimodality of solution due to high number and non-linearity of objectives.
- Multiphysics simulation model with different interacting subsystems to be integrated and automated.
1. Problem Statement
1.1 Simulation Model
A multiphysics model is developed in the COMSOL Multiphysics to simulate the behaviour of the transducer. The subsystems are the magnet and coil pair, the mechanical system with the membrane and suspension and the acoustic system with cabinet and surrounding air. The input variable is a voltage signal, the output is the sound pressure at a distance of 1 m, on-axis. A FEM of the magnet system yields the magnetic flux in the air gap, showing the stray-field as well and the decline of the flux density outside the airgap.
Figure 2. Parametrization of loudspeaker geometry
The driving force of the coil can be calculated in a lumped parameter model with non-linear elements. The coil with the metal core represents a complex impedance that varies with position. An applied voltage results in current which again results in a Lorentz force. Vice versa, the moving coil induces a current, called the back-electro-magnetic-force, resulting in a damping effect.
The mechanical properties of the membrane with suspension are modelled again in FEM. Other influences such as thermal or viscous are neglected in this study, though it is possible to include also these parameters.
1.2 Optimisation Problem Statement
The simulation model of the loudspeaker performance allows to explore various non-linear effects, including levels of distortion and compression, caused by non-linear stiffness and other non-linear parameters. Distortion and compression in the reproduced signal are estimated by the non-linear Thiele-Small (TS) parameters of the system: Bl, Le, L2, R2.
Thiele-Small parameters of the system, in fact, are functions of coil position (p_coil). In order to define the optimization targets, those curves were postprocessed to obtain the scalar measures of main acoustic features. For example, Bl curve can be qualitatively and quantitatively estimated by:
- Symmetry: 3rd central moment of distribution of the curve over zero coil position.
- Flatness: 2nd central moment of distribution of the curve over zero coil position (Variance).
- High value at zero coil position.
Le, L2 and R2 can be estimated by symmetry.
Some of those features compete and can be considered either in multi-objective optimization problem with up to 6 independent target and Pareto-frontier of optimal solutions as a result or scalarized into single objective to speed up the study.
The motor structure coupled to the mechanical system and the acoustic environment yields the non-linear behaviour at large voltage signals producing large excursions.
Figure 3. Loudspeaker at large excursion producing high sound pressure with distortion.
1.3 Optimization Routine
- COMSOL server is raised once by pSeven in the very beginning of optimization routine and being shut down in the end. This allows user to fully control execution and save overhead time on processes manipulation.
- COMSOL model being set by LiveLink connection run via user-defined Excel sheet.
- Target function calculation is done via Python in postprocessing block.
- Feasibility check is done afterwards since there are configuration of the inputs, which lead to infeasible (impossible geometry) or errors in the solver. Such configurations are handled by the optimizer as implicit constraints, allowing to continue optimization search and avoid regions of inconsistent geometries.
Figure 4. pSeven workflow for optimization problem solution.
2. Solution
2.1 DoE and Sensitivity Analysis
In many cases, especially for global exploration tasks, it is useful to perform a design of experiments (DoE) prior to run the optimization. DoE allows to examine the model behaviour the whole parameter space and estimate the impact of the parameters on the responses of the model. DoE techniques also usually computationally cheaper than Optimization techniques.
Sensitivity analysis may reveal the unimportant parameters which can be dropped from the consideration to speed up and simplify the optimization problem solution.
In Le-L2-R2 symmetry case, 2 least-important parameters were dropped off from study. The most valuable parameter variation range was narrowed closer to the desired values of target function.
Figure 5. Impact of geometry parameters on variance of the Le symmetry.
Even if all parameters are affecting the responses behaviour and the dimensionality of the problem cannot be reduced, DoE points can be effectively utilized as an initial starting sample for global search due to the nature of Surrogate-Based Optimization method applied in this problem, and the overall computation budget won’t be extended.
2.2 Optimisation strategy
In this study, we used the optimization methods based on Surrogate-Based Optimization (SBO) concept [1]. It is based on the idea of internal auxiliary approximation models, which are trained inside the optimizer and used to propose the candidates to be evaluated with a simulation model. Such internal model is retrained at each iteration using newly obtained simulation results to provide the best estimation of optimal parameters at each step.
Such approach reduces the number of simulation model evaluations. SBO algorithm supports multi-objective problem statements [2], which are important to reveal the trade-off between different configurations of the system with conflicting target functions. It also provides special means to handle implicit constraints, which may appear for certain combinations of input parameters and allows to efficiently reuse existing simulation data. SBO allows to specify the budget (number of simulation model evaluations) explicitly, which allows fitting into design schedule and obtaining global optimal solution.
3. Results
- Initial design exploration was made over scalarized target: Thiele-Small quality metrics (e.g. variance, symmetry) normalized and combined into one with user-defined weights.
- Optimization was performed within 150 simulation runs thanks to evaluation-saving nature of SBO and global optimum was revealed as a single configuration.
- Comparison of initial and optimal Le, L2 and R2 curves, all optimized for symmetry, is shown on Figure 6.
Figure 6. Impact of geometry parameters on variance of the Le symmetry.
It’s easy to notice the change in the curves shape with almost ideal symmetry of the optimal result in the whole range of coil excursion, which will result in significant reduction of loudspeaker distortion.
Conclusion
The Thiele-Small parameters of the loudspeaker were significantly improved. The result was achieved in a couple of hours where it takes days for experienced engineers to achieve a similar result. The design space was fully explored with limited number of model evaluations due to effective globalization capabilities of surrogate-based optimization algorithm, and the optimum balance of the objective functions was found.
By: Thomas Gmeiner, Director of Engineering, Mvoid Group, Anton Saratov, Vice-presidenyt of Application Engineering, pSeven SAS.
About Mvoid
Mvoid – the Virtual Product Development Expert for Automotive, Consumer and Professional Audio – is a pioneer and expert in virtual acoustics with the focus on simulation all hardware of the audio chain. The company guides its customers in Virtual Product Development by utilizing advanced Multiphysics simulations. All aspects of the audio system are modeled, culminating in real-time auralizations where the full system is listened to in validating design decisions. The Mvoid methodology enables a development environment based purely on computer-generated models and Mvoid’s VRtool.
Mvoid is an independent, owner-operated company. Mvoid offers services to the global market, retained by premium automotive, professional and consumer audio companies. The company was founded in Karlsruhe, Germany, in 2011.
Contact: thomas.gmeiner(at)mvoid-group.com
References
[1] Saw F., Fritsch T.: "Title", John Wiley & Sons, 1995.
[2] Forrester A, Keane A. Recent advances in surrogate-based optimization, Progress in aerospace sciences. –Elsevier, Southhampton, UK . – 2009. P. 1 – 77.